Coordinate independent expression for transverse trace-free tensors

TL;DR

This paper presents a coordinate-independent formulation for transverse trace-free tensors in flat and conformally-flat spaces, extending previous symmetric cases to more general invariance conditions relevant for numerical relativity.

Contribution

It introduces a coordinate-free expression for TT tensors, generalizing symmetry conditions to invariance along any hypersurface orthogonal Killing vector, enhancing their applicability in gravitational initial data.

Findings

Provides a coordinate-independent formula for TT tensors in flat space.

Extends symmetry conditions to invariance along arbitrary hypersurface orthogonal Killing vectors.

Facilitates the use of TT tensors in numerical relativity with more general symmetry assumptions.

Abstract

The transverse and trace-free (TT) part of the extrinsic curvature represents half of the dynamical degrees of freedom of the gravitational field in the $3 + 1$ formalism. As such, it is part of the freely specifiable initial data for numerical relativity. Though TT tensors in 3-space possess only two component degrees of freedom, they cannot ordinarily be given solely by two scalar potentials. Such expressions have been derived however, in coordinate form, for all TT tensors in flat space which are also translationally or axially symmetric (Conboye and \'O Murchadha 2014 \emph{Class. Quantum Grav.} {\bf 31} 085019). Since TT tensors are conformally covariant, these also give TT tensors in conformally-flat space. In this article, the work above has been extended by giving a coordinate-\emph{independent} expression for these TT tensors. The translational and axial symmetry conditions have also been generalized to invariance along any hypersurface orthogonal Killing vector.